p. 4

NJ:

and

ont,

5.A A RATIONALE FOR MULTIOBJECTIVE DECISION MODELS

For many engineering management problems, particularly those in the public sector,more than one objective is generally important. Consider the problem of developingan operating strategy for a large, multipurpose water reservoir. It is not uncommonfor such a facility to be used to meet a variety of societal needs including municipalwater supply, agricultural water supply, flood control and streamfiow management,hydroelectric power production, outdoor recreation, and protection of fragile environmental habitat. From a management perspective, these uses of such a facilityconflict with one another. For example, an optimal management strategy with respect to ensuring a reliable supply of water from a reservoir for municipal use mighthave as its objective function: maximize storage in the reservoir at all times. Yet forpurposes of containing an extreme flood event, the objective function could be justthe opposite: minimize storage in the reservoir at all times. Several optimization techniques have been developed to capture explicitly the tradeoffs that may exist between conflicting, and possibly noncommensurate, objectives. In this chapter we laythe foundation for multiobjective analysis that can be applied across the spectrumof public sector engineering management problems and demonstrate the application of two of the most useful multiobjective optimization methodologies.

Multiobjective programming deals with optimization problems with two ormore objective functions. The multiobjective programming formulation differs fromthe classical (single-objective) optimization problem only in the expression of their

Linear Programs withMultiple Objectives

121

122 Linear Programs with Multiple Objectives Chap.

respective objective functions; the multiobjective formulation accommodates explicitly more than one. Yet the evaluation of management solutions is significantly different: instead of seeking an optimal or best overall solution, the goal of multiobjectivanalysis is to quantify the degree of conflict, or tradeoff among objectives. From another perspective, we seek to find the set of solutions for which we can demonstratthat no better solutions exist. This best available set of solutions is referred to as thset of noninferior solutions. It is from this set that the person or persons responsiblfor decision making should choose; the role of the systems analyst is to describe as accurately and completely as possible the range for that choice and the tradeoffs amonobjectives between members of that set of management solutions. Noninferiority ithe metric by which we include or exclude solutions in this set.

5.A.1 A Definition of Noninferiority

In single-objective problems our goal is to find the single feasible solution that provides the optimal value of the objective function. Even in cases where alternate optima exist, the optimal value of the objective function is the same for each alternatoptima (extreme point), as can be seen in Figure 3.3b. For problems (models) havinmultiple objectives, the solution that optimizes any one objective will not, in gener-,al, optimize any other. In fact, for decision-making problems that are most challenging from an engineering management perspective, there is usually a very largdegree of conflict between objectives such as in the example of reservoir management. Another example might be in the area of structural design, where objectivesmight include the maximization of strength concurrent with the desire to minimizeweight or cost. In managing environmental resources, we might seek to trade off environmental quality and economic efficiency concerns, or even conflicting environmental quality goals; minimizing the volume of landfill disposal against dischargesto the atmosphere by incineration of municipal refuse. Note that in these latter examples, the units of measure for different objective functions may be quite differentas well. We call such objective functions noncommensurate.

When dealing with objectives that are in conflict, the concept of optimalitymay be inappropriate; a strategy that is optimal with respect to one objective maylikely be clearly inferior for another. Consequently, a new concept is introduced bywhich we can measure solutions against multiple, conflicting, and even noncommensurate objectives—the concept of noninferiority:

A solution to a problem having multiple and conflicting objectives is noninferiorif there exists no other feasible solution with better performance with respect to any oneobjective, without having worse performance in at least one other objective.

Noninferiority is similar to the economic concept of dominance and is evencalled nondominance by some mathematical programmers, efficiency by statisticians and economists, and Pareto optimality by welfare economists. A simple extension to the Homewood Masonry problem presented as Example 3-1 will help thereader understand this often nebulous concept.

Sec. 5.A A Rationale for Multiobjective Decision Models

5.A.2 Example 5-1: Environmental Concerns for Homewood Masonry

123

Problem Statement. The management of Homewood Masonry has longbeen concerned about the local environmental impacts of their production operation.both as a responsible member of the community within which their plant resides, andin anticipation of increasingly tighter standards and governmental controls. You havebeen asked to study the operation of the plant and to identify from a technical perspective the level of conflict that exists between these two management obiectives.

After analyzing the results of a comprehensive air-monitoring program, youdiscover that the major environmental impact of the operation results from a release of contaminated dust during the blending process; the binder used in manufacturing both HYDIT and FILIT attaches to these dust particles, and is therebyreleased to the environment during production. Laboratory tests suggest the releaseof this pollutant from the plant amounts to 500 milligrams for each ton of HYD1Tproduced and 200 milligrams for each ton of FILIT produced. A second objectivefunction, one that seeks to minimize total plant emissions, can now be specified as

Minimize Z2 = 500x1 + 200x2.

The feasibility of solutions (the feasible region in decision space) is not affected bythe consideration of this objective function. Note that the sense of this objectivefunction is opposite that of our original production objective function (maximize total weekly revenue), and the units (milligrams discharged) are different as well(dollars). Yet both objective functions are related through the same set of decisionvariables.

By the same argument presented in the previous chapter, the solution that optimizes this second objective function must be a basic feasible (extreme point) solution in the original problem. The values for the decision variables at each of thesesolutions are repeated in Table 5.1, and the values of both objective functions ateach of those solutions are included. Not surprisingly, the solution that optimizes theenvironmental objective is the “do nothing” solution: x1 = 0. x = 0.

TABLE 5.1 DECISION VARIABLES AND THEIR VALUES FOR ALL FEASIBLE EXTREME POINTSOLUTIONS FOR THE TWO-OBJECTIVE HOMEWOOD MASONRY PROBLEM: EXAMPLE 5-1

Alternative r Max Z1 Mm Z Noninferiority

A 0 0 0 0 NoninferiorB 8 0 1120 4000 Dominated by D. E

C 8 2 1440 4100 Dominated by D

D 6 4 1480 3800 Noninferior

E 2 6 1240 2200 Noninferior

F 0 6 960 1200 Noninferior

124 Linear Programs with Multiple Objectives Chap. 5

Because both objective functions have been previously specified, and are thus

assumed to reflect the overall goals of production and environmental concern (im

plicitly, we assume that there are no other management objectives), we can apply

the concept of noninferiority as defined above to each of these solutions (produc

tion alternatives). Notice that the solutions that optimize the individual objective

functions Z1 andZ2—alternative A and alternative D, respectively—are indicated

as being noninferior. In fact, for any multiobjective optimization model, the solution

that optimizes any single objective function is always noninferior, unless there are

alternate optima at that solution with respect to that objective function (this qualifi

cation will be clarified later). By the definition of noninferiority, if a solution is opti

mal for a given objective function, it is not possible to find a clearly better feasible

solution regardless of how that solution might perform with respect to any (or all)

other objective functions.

Consider alternative B. It is not the worst solution with respect to profit; it is

clearly better than alternative F by this measure. Nor is it the worst solution envi

ronmentally; it is better than alternative C. But given the stated objective functions

and awareness of this set of alternatives, would you ever select alternative B? Would

anybody ever select alternative B? Stated another way, is there any alternative that

would always be preferred to alternative B by anybody having preferences repre

sented by this specific set of objective functions? The answer, of course, is that both

alternatives D and E perform better with respect to both objectives than does alter

native B, so that no decision maker would ever implement alternative B if he or she

were aware of the availability of alternatives D or E. Similarly, alternative C is clearly

dominated by alternative D.

Now let’s compare alternative D—an alternative that has already been shown

to be noninferior—with alternative E. While alternative D represents a production

strategy that maximizes profit for Homewood Masonry, it would also have a more

adverse impact on the local environment than would E. Therefore, the choice be

tween these alternatives is not obvious, and probably depends on the specific pref

erences of the decision maker. It is easy to envision a scenario in which the board of

directors of Homewood Masonry might themselves be divided over which strategy

to implement, particularly if they reside in the vicinity of the plant, for instance. Can

you see that the same logic applies to the determination that alternative F is also

noninferior?The goal of such an analysis is thus to identiy all solutiqg that are ioninfrri

or: the set of solutions for which there does not exist añothcr soluti9ri thaLwould al

ways be preferable to any of those solutions. This set of alternatives is referred to as

the noninferior set, or sometimes the Pareto frontier. It is then the responsibility of

the decision maker to select from among these solutions that which represents their

best compromise solution among the stated objectives.

The definition of noninferiority seems more difficult to state than to compre

hend. Make sure you understand the logic used to determine dominance and non

dominance with respect to the solutions presented in Table 5.1, then review carefully

the definition of noninferiority given above. When you feel that you’ve mastered the

Sec. 5.A A Rationale for Multiobjective Decision Models 125

TABLE 5.2 DECISION VARIABLES AND THEIR VALUES FOR ALL FEASIBLE EXTREME POINTSOLUTIONS FOR A MORE COMPLICATED THREE-OBJECTIVE PROBLEM: EXAMPLE 5-2

Alternative x1 Max Z1 Max Z2 Mm Z3 Noninferiority

A 2 0 6 —2 2 NoninferiorB 4 1 10 —2 5 NoninferiorC 6 5 8 4 11 NoninferiorD 6 7 4 8 13 NoninferiorE 3 6 —3 9 5 NoninferiorF 1 5 —7 9 6 Dominated by EG 0 3 —6 6 12 Dominated by EH 0 1 —2 2 1 Noninferior

concept, examine the data for a three-objective, eight-solution multiobjective program presented in Table 5.2 and try to verify that the noninferior set for this problem consists of points A, B, C, D, E, H. Note that objective Z2 has alternateoptima—both solution E and solution F provide an objective function value of 9—but for the three-objective problem, solution E dominates solution F

You might also try writing your own objective function that depends on thosevalues of the decision variables x1 and x2, and see how the inclusion of this fourthobjective function changes the noninferior set. You should start to realize that thedetermination of noninferiority gets increasingly complicated as the problem growsinslin both number of basic feasible extreme point solutions and number of objective functions. Most real engineering problems in the public sector have hundredsof thousands of feasible extreme points, and may have tens of objective functions.Before we discuss a general-purpose algorithm for identifying the noninferior set, itis useful to develop a graphical framework within which to study further the conceptof noninferiority.

5.A.3 A Graphical Interpretation of Noninferiority

Consider the two-objective mathematical program presented below:

Maximize [Zi(xi, x7),Z2(xi, 17)]

where: Z1 = 3x — 212

Z2 = —x1 + 2x2

Subject to: 4x1 + 817 8

3x1 612 6

4x1 — 2x2 S 14

126 Linear Programs with Multiple Objectives Chap. 5

—x1 + 3x2 15

—2x1 + 4x2 18

—6x1 + 3x2 9

x1, x2 0.

The feasible region in decision space and the objective functions are plotted in

Figure 5.1, with each basic feasible solution labeled A—H.

The most astute of readers will have noticed that this two-objective problem us

es the same feasible region specified in Table 5.2 as well as the first two objective func

tions listed in that table (we will ignore the third minimization objective for the time

being). The shaded cells in that table indicate the optimal solutions. The presence of

alternate optima for Z2 is not surprising if we note that the coefficients that multiply

the decision variables in that objective (—xi + 2x2) result in an objective function

having a slope that is identical to one of the binding constraints (—2x1 + 4x2 18).

Because we have limited our example problem to not more than three objec

tives, we can map the feasible region in decision space to a corresponding feasible

region in objective space; we simply plot the ordered (Z1,Z2) pairs as presented in

Figure 5.2. Using the common reference provided by Table 5.2, each basic feasible

solution labeled in Figure 5.1 has a corresponding solution in objective space using

the same letter designator. For example, point B in Figure 5.1 corresponds to Point

B in Figure 5.2, with the corresponding coordinates taken from Table 5.2. Signifi

cantly, adjacent feasible extreme points in decision space map to adjacent solutions

in objective space. Whereas the shape of the feasible region in decision space de

pends on the constraint set for a particular problem, the shape of the feasible region

7

6 -

4

- zI

Figure 5.1 The feasible region in decision I

space for the problem presented in Table 5.2with solutions that optimize Z1 and Z2 shownpassing through their respective optima— 1 2 4 5 6 7

points B and F, respectively. Xj

Sec. 5.A A Ralionale for Multiobjective Decision Models 127

Figure 5.2 The feasible region inobjective space is defined by plotting allbasic feasible solutions from decisionspace mapped through the objectivefunctions Z1 and Z2. Noninferiority is theneasily determined using the northeastcorner rule.

in objective space depends on the objective functions, which serve as “mappingfunctions” for a particular set of objectives.

This graphical representation provides a much easier means for identifyingnoninferior solutions. First, it should be obvious that all interior points must be inferior, because given any such point, one would always be able to find another feasiblesolution that would improve both objectives simultaneously. For example, considerinterior point P in Figure 5.2, which is inferior. Alternative D gives more Z1 thandoes P without decreasing the amount of 4. Similarly, D gives more Z2 without decreasing Z1. In fact, any alternative in the shaded wedge shape to the “northeast” ofpoint P dominates alternative P. We can generalize this notion in the form of a rulehaving this directional analog:

A feasible solution to a two-objective optimization problem in which both objective functions are to be maximized is noninferior if there does not exist a feasiblesolution in the northeast corner of a quadrant centered at that point.

Applying this northeast corner rule to the rest of the entire feasible region inFigure 5.2 leads to the conclusion that any point that is not on the northeastern boundary of the feasible region is inferior. The noninferior solutions for the feasible region inFigure 5.2 are found in the thickened portion of the boundary between points B and E.Use the northeast corner rule to convince yourself that solution F is indeed dominated(by solution E) and is thus not a member of the noninferior set even though it wasshown to be an alternate optimum when we solved Z2 as a single-objective optimization.

We can, of course, generalize this result to evaluating solutions for problemswith any combination of objective function sense. For example, in the current problem if, instead of both objectives being maximized, they were minimized. Can yousee that the noninferior set would then consist of those solutions on the southwestborder of the feasible region in objective space between points A and F? What ifone objective is a maximization and one a minimization? What should you conclude if the tradeoff surface (noninferior set in objective space) reduces to a singlepoint?

Northeastcorner

Noninferior setin objective space

NoA B

f

yEl

eneg

I—

[5

n

128 Linear Programs with Multiple Objectives Chap. 5

The noninferior set for the two-objective problem that we just solved consistedof the points labeled B, C, D, and E. Yet when considering a third objective Z3 inTable 5.2, points A and H are also included in the noninferior set. An important assumption underlying multiobjective analyses is that the decision maker(s) must beable to articulate all relevant objectives for a particular problem. Otherwise, solutions that are noninferior may be excluded from consideration in the same way thatfor our hypothetical example, we would not consider alternative H for implementation without a consideration of objective Z3.

Now that we are comfortable with the concept of noninferiority, let’s examinetwo methodologies that will allow us to identify efficient solutions when it is notpossible to graph our solution space. We will demonstrate these techniques with thesample problem we just studied, but the reader should appreciate that the methodologies are applicable to any multiobjective model.

5.B METHODS FOR GENERATING THE NONINFERIOR SET

A number of methodologies have been devised to portray the noninferior setamong conflicting objectives. We will confine our treatment of this topic to a class oftechniques that enjoys widespread use among engineers. Generating techniques, asthey are commonly called, do not require (or allow) decision makers’ preferences tobe incorporated into the solution process. The relative importance of one objectivein comparison to another is not considered when identifying the noninferior set, butused later on to compare noninferior solutions and to quantify the tradeoffs between them. Typically, analyst(s) will work iteratively with the decision maker(s) toidentify a complete set of objective functions for a particular problem domain andto specify the appropriate set of decision variables to relate these objectives to oneanother and to problem constraint conditions. The noninferior set is then generatedby the appropriate technique, such as those presented below, and presented to thedecision maker for further consideration.

The selection of a solution to be implemented from among those solutions inthe noninferior set is the responsibility of the decision maker(s). The strength of theuse of generating methods for multiobjective optimization is that the roles of the analyst(s) versus the decision maker(s) are as they should be: the analyst providescomprehensive information about the best available choices in a given problem domain, and the decision maker assumes the responsibility for selecting among thosechoices. The analyst is not involved with making value judgments about the relativeimportance of one objective over another, and the decision maker need not worryabout the technical aspects of the physical system nor fear that better solutions arebeing overlooked.

We will present two methods for generating the noninferior set: the weightingmethod and the constraint method. There are strengths and weaknesses of eachmethod for a given application, but they both rely on the repeated solution of linearprograms. The general single-objective optimization with n decision variables and m

Sec. 5.B Methods for Generating the Noninferior Set 129

constraints was presented in Chapter 2 (Section 2.B). The general multiobjectiveoptimization problem with n decision variables, m constraints, and p objectives is:

Optimize Z Z1 (x1, x2 x,1)

Z2(x1,x2 x)

Zp (xi, 2,”,n)

Subject to: g1 (x1. x2 ) b1

g2(xi,x2 x) b

x) brn

0V1 (for allj).

where Z(xi,x7,...,x) is the multiobjective objective function and Z1(j. Z2Q.Z1)() are the p individual objective functions. Note that the individual objective

functions are merely listed; they are not added, multiplied, or combined in any way.For convenience of illustration, we will assume that all objectives in the model arebeing maximized.

5.B.1 The Weighting Method of Multiobjective OptimizationThe weighting method is acknowledged as being the oldest and probably most frequently used multiobjective solution technique. Once the objectives, decision variables, and constraint equations have been fully specified, the weighting method canbe accomplished as follows:

1. Solve p linear programs, each having a different objective function. Each ofthese solutions is a noninferior solution for the p objective function problemprovided that alternate optima do not exist at that solution. If alternate optimaare indicated, at least one of the optimal basic feasible extreme points will benoninferior (it is possible that more than one will be noninferior, but likelythat some will be dominated).

2. Combine all objective functions into a single-objective function by multiplyingeach objective function by a weight and adding them together such that

Maximize Z = 7, Z

becomes

Maximize Z(wi, ‘2 w) = w1Z1 + w2Z, + ... +

This objective function is often referred to as the grand objective.

130 Linear Programs with Multiple Objectives Chap. 5

3. Solve a series of linear programs using the grand objective while systematicallyvarying the weights on the individual objectives. Each of these solutions willbe a noninferior solution for the multiobjective problem. The number of different sets of weights and the number of linear programs solved depend on the icomplexity of the tradeoff surface and the time available to the analyst.

The weighting method will be used to solve the two-objective problem that wassolved graphically (and exhaustively) in Section 5.A.3.

Solve p Individual Linear Programs. Solving our model Z1 as the only objective function may be viewed as moving a vertical line through objective spacesimilar to how we solved graphical problems for which we could plot decision space.The feasible region for objective space for the current problem is reproduced inFigure 5.3, with the objective function gradient for Z1 shown as the vertical linepassing through point B—the optimal solution for that single-objective problem. Werefer to this gradient of the objective function as gradient 1.

The same procedure is then used to solve the single-objective problem usingZ2, which is also plotted on Figure 5.3—the horizontal line labeled gradient 2. Recall from our previous experience that alternate optima exist for this solution—points E and F Analytical procedures for determining which of these optimalsolutions is noninferior will be presented later.

Set Up the Grand Objective Function. The grand objective function isformed by multiplying each objective function by a weighting factor and addingthese weighted objective function terms together. For our problem, the grand objective function is written

Figure 5.3 Optimizing individualobjective problems may be viewedas moving that objective functionthrough objective space with aweight of one on that objective,and a weight of zero on all otherobjective functions. Changing theweights on objectives changes thegradient of the grand objectivefunction.

2 II

+

N

N

E

z1

Maximize ZG w1Z1 + x2Z2

= wi(3x1 — 2x2) + w2(—xi + 2x2).

Maximize 0Z1 + 1Z2 9

L98

—5

Sec. 5.B Methods for Generating the Noninferior Set 131

For minimization objectives, one can multiply that objective function by —1 tochange its sense to a maximization, The weight is a variable whose value will changedsystematically during the solution process. It will be clear as we begin to generatenoninferior solutions that it is the relatit’e weights on these objective functions that isimportant, not their specific values.

Generating the Noninferior Set. For each set of positive weights used inthe grand objective function, the resulting solution will be a noninferior solution.Grand objective functions for sets of weights are shown in Table 5.3. For example.suppose we set weights of 0.9 for w1 and 0.1 for w in the grand objective functionthat we just constructed. The resulting single-objective function in solving a normallinear program would be

Maximize ZG= O.9(31i — 2x2) + O.1(—xi + 2x)

= 2.6x— 1.612.

This objective function is labeled as gradient 3 in Table 5.3 and as plotted on Figure 5.3.Gradients 1 and 2 were those that resulted from optimizing each objective function individually and are also included in that table and plotted in objective space. We complete the table by ranging the weights w1 and w2 from 1 to 0 and 0 to 1, respectively, andplotting these gradients in objective space as well.1

TABLE 5.3 THE GRAND OBJECTIVE FUNCTION AND THE CORRESPONDING WEIGHTS ON THEINDIVIDUAL OBJECTIVES ARE SHOWN TOGETHER WITH THE NONINFERIOR SOLUTION THATWOULD RESULT FROM OPTIMIZING THIS OBJECTIVE FUNCTION. THE GRADIENT NUMBERREFERENCES THE CORRESPONDING OBJECTIVE FUNCTION IN FIGURES 5.3 AND 5.4

Gradient w2 Objective Function ZG Solution

1 1.0 0.0 3x1 2x2 B

3 0.9 0.1 2.6x1 — 1.6x2 B

4 0.8 0.2 2.2x 1.2x, B

5 0.7 0.3 1.$x 0.8x C

6 0.6 0.4 1.4x1 — 0.4x C

7 0.5 0.5 xl C, D

8 0.4 0.6 0.6x1 + 0.4x2 D

9 0.3 0.7 0.2x1 + 0.8x2 D

10 0.2 0.8 —0.2.s + 1.2x2 D

11 0.1 0.9 —0.6x1 + I 6x E

2 0.0 1.0 —x. + 0.2x F.E

lit is the relative values of these weights that is important; the convention that the weights sum toone is a convenience,

132 Linear Programs with MutipIe Objectives Chap

Figure 5.4 The feasible region in decisionspace for the problem presented in Table5.2 with solutions that optimize Z1 and Z2shown passing through their respectiveoptima—points B and F—F, respectively, andgradients in decision space for the grandobjective function shown in Table 5.3.

I

Note that each time we solve a linear program using the resulting grand objectivfunction, the solution is a noninferior feasible extreme point. Because it is importadto recognize the relationship between objective space and decision space as we gererate the noninferior set, we also reproduce the graph of the feasible region in decsion for this problem as Figure 5.4 and plot gradients 1 through 11.

Care when selecting a strategy for varying these weights is essential. In solvithis problem using the weighting method we had the advantage of knowing what thnoninferior set was because we had a graphical representation of the solution spacThe convention that the weights sum to one was not necessary, but is common praøtice. The important thing is to develop a procedure having fine enough resolution sthat all solutions can be found. In this case we could have incremented/decrementethe values of the weights by 0.2 instead of 0.1 and still have found all noninferior sclutions with half as much computational effort. For relatively flat portions oftradeoff curve, a smaller increment for relative weights may be necessary. Andthe number of objective functions increases, the number of combinations of weighon those objectives in the grand objective function increases as well. For largproblems, the analyst must always be concerned about balancing the computationeffort required to find all noninferior solutions, which may be prohibitive, againthe necessity of finding all noninferior solutions. In general, however, computatiolis very cheap when measured against the value of being able to hand a decisimaker a true and complete noninferior set of management alternatives.

5.B.2 Dealing with Alternate Optima When Using the Weighting Method

While performing the weighting method of generating the noninferior set formultiobjective optimization having p individual objectives, alternate optima ma

7—

6—

5—

21 NIII

5 6 72M4i3 xl

Sec. 5.B Methods for Generating the Noninferior Set 133

be encountered (1) when solving one or more problems using the individual p objectives or (2) when solving a linear program using the grand objective functionconstructed by weighting and combining all p objective functions. In the firstcase, the analyst must realize that only one of the alternate optima is a noninferiorsolution as was shown by using the northeast corner rule on Figure 5.2. In the second case, all alternate optima are also noninferior; notice, for example, that gradient 7 in Table 5.3 and Figure 5.3 found noninferior points D and C, which wouldhave been indicated as alternate optima when solving the grand objective function indicated.How can we determine which alternate optima are noninferior if such a condition is detected when solving the individual p optimizations? Recall that when wesolved the linear program using only objective function Z2 in the previous examplealternate optima were indicated—points F and Fin Figure 5.3 and Table 5.3. for gradient 2. This model is the same as one having a weight of 1 on objective Z2 and aweight of 0 on objective Z1. What if. instead, we had chosen to solve the problemwith a weight of 1 on objective Z1 and a weight of E on Z1:

Maximize 7, = eZ1 + 1.07,

where s is an infinitesimal positive weight on Z1. This would have the effect oftilting gradient 2 in Figure 5.2 slightly clockwise such that the optimal solution tothis modified problem would be the single feasible extreme point E—the extremepoint that is noninferior for the multiple Objective model when optimizing Z2 byitself.This method can be generalized for problems of any size having any numberof objective functions as follows:For any multiobjective problem having p objective functions, if the solution obtained when solving the ith objective function displays alternate optimal solutions thenoninferior alternate optima will be that solution resulting from solving a linear program having a grand objective function with a weight of] on the ith objective, and aweight of s on the remaining p — I objectives.Of course one should not be surprised if the solution obtained after solvingthis second model is identical to the first solution. It is just that this time the solution will be a unique optima. Another method for finding the noninferior alternateoptima will be discussed later and should serve to strengthen your understandingof this concept.

5.B.3 The Constraint Method of Multiobjective Optimization

An alternate method for generating the noninferior set in objective space having pobjectives is called the constraint method. After solving p individual models to identify the solution that optimizes each, one objective function is selected (arbitrarily)to be optimized, with the other objective functions included in the constraint set

134 Linear Programs with Multiple Objectives Chap.

with right-hand sides set so as to restrain the value of the objective function thaiwas selected for optimization. By iteratively solving this modified formulation, andbecause, as with the weighting method, each solution to the modified problem is

noninferior solution to the original problem, an approximation of the noninferio

set in objective space can be generated. The same hypothetical two-objective prob.

lem presented in Section 5.A.3 will be solved below using the constraint method

Construct the Payoff Table. Solve p individual optimization problems anconstruct a payoff table, shown as Table 5.4. The payoff table is a p X p matrix wi

a column for each objective function, and a row for each optimal solution. For example, solution vector x1 is the solution that optimizes Z1 with a value of 10; at thj

solution, the value for objective function Z2 is —2. Each solution listed in the payo

table must be noninferior. If alternate optima are detected when solving any of th

p individual formulations—as in this case when we solved Z2 (see Figure 5.2)—th

noninferior alternate optimal solution must be determined. Consider the formula

tion that optimizes Z2 for this sample problem:

Maximize Z2 = —x1 + 2x2

Subject to: 4x1 + 8x2 8

3x1 — 6x2 6

4x1 — 2x2 14

x1

—x1+3x215

— 2x1 + 4x2 18

— 6x1 + 3x2 9

xi, x2 0

which gave a solution x1 = 1. x2 = 5, and Z 9, with alternate optima indicated.

An alternate method for determining which of these alternate optima is noninferi

or is to modify this formulation so that Z2 is constrained to a value of 9, while Z1 is

optimized:

TABLE 5.4 PAYOFF TABLE FOR A HYPOTHETICAL TWO-OBJECTIVE OPTIMIZATION PROBLEM

Extreme Point

Solution x1 x2 = 3x1 — 2x2 Z2 = —x1 + 2x2 (Figures 5.2 & 5.3)

x’ 4 1 10 —2 B

x2 3 6 —3 9 E

II

Sec. 5.B Methods for Generating the Noninferior Set 135

t Maximize Z1 = 3x1—

Subject to: 4x1 + 8x2 8

r 3x1—6x26

4x1—2x214

—x1+3x215

—2x1 + 4x2 18

—6x1+3x29ee —x1+2x29

x1,x20.

The optimal solution to this subproblem will always be the noninferior optimal solution to the single-objective problem. When there are more than two objectivefunctions being modeled, this subproblem can have as its objective function the total of all p — 1 other objective functions.

The significance of the payoff table is that it identifies, for each objective function, the range of values each can have on the noninferior set. This range is boundedabove by the largest value in the payoff table corresponding to that objective function and below by the smallest value. In our example, there will not exist a noninfenor solution in Z1, Z2 space that has a larger value for Z1 than 10, nor a smallervalue for Z1 than —3:

Lax 10 =

L2 —o 12 ——

max — mill —

For problems having more objectives, the size of the payoff table is larger, but therange for each objective is determined by these limits.

Set Up the Constrained Problem. The constrained problem is specified byselecting one objective function (arbitrarily) for optimization, and moving all otherp — 1 objectives into the constraint set with the addition of a right-hand side coefficient for each. This coefficient will be between Lmax and Lmin for all objective functions. Selecting Z1 to optimize and moving Z2 into the constraint set gives theconstrained problem

Maximize Z1 3x1—

Subject to: 4x1 + 8x2 8

136 Linear Programs with Mu$tiple Objectives Chap.

3x1 — 6x7 6

4x1 — 2x2 14

x1

—x1 + 3x2 15

—2x1 + 4x2 18

—6x1 + 3x2 9

—x1 + 2x2 L with L

x1,x2 0.

Generate an Approximation of the Noninferior Set. By repeatedlysolving the constrained problem, an approximation of the full and precise noninfenor set in objective space can be generated. As with the weighting method, the optimal solution to each constrained problem is a noninferior solution to the originalproblem.

The precision with which one approximates the true noninferior set using theconstant method depends on the number of times one is willing or able to solve theconstrained problem developed in step 2. Let r be the number of noninferior solutions to be generated in such an approximation; for this example, r = 5. Then the (5)values for the right-hand side of objective Z2 in the constraint set are determined using the following formula:

Lk Lmin +[(r l)1(Lrnax — Lmin), fort 0, 1,2 (r — 1).

Solving this equation five times results in five different values for Lk:

L1 = -2 + [][9 - (-2)j = -2

L2 —2 + [i1 — (—2)] = 0.75

= -2+

[][9 - (-2)] = 3.5

= -2 + [][9 - (-2)] = 6.25

L5 = -2 + [][9-

(-2)] = 9.

Sec. 5.B Methods for Generating the Noninferior Set

1/ 1- I-1 2 3 4 5

8.2 z, L = 3.5

Figure 5.5 The feasible region in2 decision space for the example

problem shown with Z2 constrainedto five different values of Lk, and thecorresponding location of theobjective function Z1. Thecorresponding values of Z1 and Z,indicate noninferior solutions.

Solving the constrained problem five times using a different value of Lk each timewill result in a noninferior set of five solutions evenly distributed across the Z2 axisbetween Z2 Lmin and Z2 = Lmax, inclusive. Figure 5.5 shows the feasible region indecision space with each of these new constraints plotted as a dashed line, and theoptimal gradient Z1 when that particular problem is solved. The corresponding so!id points on the graph represent the resulting noninferior solutions in decisionspace, numbered 1 through 5. These solutions are summarized in Table 5.5. Notethat in all cases the value for Z2 at that noninferior solution is equal to the value forLk, suggesting that constraint—objective Z2 in the constraint set—is binding. Thecorresponding solution space in objective space is presented as Figure 5.6. The non-inferior solutions generated by the constraint method are included as solid dots.

TABLE 5.5 SUMMARY OF NONINFERIOR SOLUTIONS GENERATED BY THE CONSTRAINT METHOD

Z=—3 Z2L=9Z15.73

L. 6.25

137

= 9.1

6 7xI

NoninferiorSolution Lk x1 Z1 = 3x1 — 2x2 Z2 = —x1 + 2x2

1 —2 4 1 10 —2

2 0.75 4.9 2.83 9.1 0.75

3 3.5 5.82 4.66 8.2 3.5

4 6.25 6 6.125 5.75 6.25

5 9 3 6 —3 9

138 Linear Programs with Multiple Objectives Ch

4 = -34=8.2

I z2 4=5.75 4=9.1

i4

- --A---- 4=

A 4=-5

b---

Again, the objective functions Z1 and Z2 are shown passing through each noninfor solution, labeled 1 through 5. The heavy solid line connecting these points isapproximation of the noninferior set.

Notice that the approximation to the noninferior set generated by thestraint method does not “find” noninferior solutions C and D, which were founding the weighting method. Furthermore, the approximation is better in the rebetween points B and C than it is between points D and E. By increasing the nber of points used to approximate the noninferior solution, however, we can geclose an approximation as we wish. Typically, after an initial screening of solutithat are evenly distributed within the range Lmin and Lmax for each objective ftion shifted to the constraint set, additional values can be used to search for noferior solutions in regions of the solution space that might be more important todecision maker.

5.B.4 Selecting a Generating Method

Two of the more important and popular methods for generating the noninferiorin objective space between two or more conflicting objectives are the weightingthe constraint methods presented in this chapter. Other methods have beenposed, and the selection of the best one for a particular analysis depends, formost part on the experience of the analyst.

As the number of objectives and size of the solution space increases, theputational effort required to find all noninferior solutions rises dramatically.methods rely on the judgment of the analyst as to the configuration and shape ofnoninferior set.

When the degree of conflict between objectives is expected to be significthe weighting method is quite effective at finding noninferior solutions. But ifcoarse a resolution of weights for the weighting method is specified, noninferiorlutions may be missed. In fact it is possible that noninferior solutions exist

Figure 5.6 The feasible region inobjective space for the example problemshown with Z2 constrained to fivedifferent values of Lk, and thecorresponding location of the objectivefunction Z1. The heavy black lineconnecting points 1 through 5 is theapproximation of the noninferior set.

Chap. 5 Chapter Summary 139

would not be found with any combination of weights.2 If too fine a resolution isspecified, any combinations of weights may find the same noninferior solution andcomputation costs may be excessive.

If the decision maker is able to articulate a preference for one or more objectives in terms of a range of acceptable values, then the constraint method may bemore effective because these objectives can be moved to the constraint set withright-hand side values that are specifically, rather than generally, specified to best“cover” a region of greatest interest for the decision maker. For example, if the decision maker has articulated a concern that costs be minimized, he/she may be further encouraged to specify an acceptable region for these costs: “Cost should beminimized, but in no case be allowed to exceed X dollars.”

For analysts working in the public sector, virtually every decision, and therefore every model constructed to support those decisions, has a multiobjective context. It is incumbent upon the analyst to explore with the decision maker the entireframework within which management decisions are made. Frequently, decisionmakers do not appreciate the efficiency of the analytical tools available to addressexplicitly the tradeoffs between objectives and may not even be aware of all the objectives that should or could be considered related to a specific decision. The notionof noninferiority as it pertains to public sector decisions and the ability to generatethe noninferior set among conflicting objectives are among the most powerful toolsof the analyst who works on public sector problems.

CHAPTER SUMMARY

More frequently than not, management decisions in the public sector must considermultiple and often conflicting objectives. When this is the case, there may not be optimal solutions. Rather, the solution that optimizes one objective will not optimize aconflicting objective function. The least-cost solution may not be one providing thebest or most reliable level of service for example Instead of trying to find an opti

mal solution, we are now interested in finding the full set of noninferior solutions—asolution is noninferior if there does not exist another solution that performs betterin terms of one objective function without performing worse with respect to at leastoneherThd complete set of noninferior solutions is called the noninferior set anddefines the trade ffs bfn objectives

Athodification to the basic linear programming formulation, with a corresponding modification to the way in which the model is solved, can define the precise nature of the tradeoffs that exists between such conflicting objectives, thereby

2Suppose a solution Z1 400, Z2 1000 was added to the set of solutions presented in Table 5.1.This solution would be noninferior even though no combination of weights for the two-objective functions would find it in objective space. Plot these solutions and convince yourself of this fact. These solutions are ca]led gap point solutions and are quite common when the number of objectives beingconsidered is large, and the solution space for one or more objectives is discrete.

140 Linear Programs with Muhipe Objectives Chap. 5

generating the noninferior set in objective space. Two reliable methods for solvingsuch problems of multiobjective optimization are the constraint method and theweighting method. Both methods begin by computing the solutions that optimizeeach individual objective being modeled.jescso1utions are unique optirnhey

<jonjnferior. If not, exactly one of each alternate optima is noninferior for thatobjective.

The constraint method proceeds by transforming the original multiobjectiveproblem formulation into a single-objective optimization problem by placing all except one objective function into the constraint set, with the right-hand side of eachset within a range of values limited above by its value when the corresponding single-objective problem was solved, and below by the worst of its values from the othersingle-objective optimizations. By ranging the right-hand sides for all such constraints in the transformed problem and resolving each new problem, the noninferior set is generated; each new solution is a noninferior solution. The weightingmethod is similar, but the objectives are added to make up a grand objective function, with weights assigned to each.

EXERCISES

5.1. Five different late-night telemarketing programs guarantee different quantities of love,money, health, fame, and friendship—for different monthly payments, of course—as indicated by the table below:

Program Program Program Program ProgramThepromise A B C D E

Love 2 2 5 1 2

Money 3 4 4 2 2

Health 3 2 4 2 2

Fame 1 2 0 1 2

Friendship 2 5 1 5 2

Low Monthly Cost $29.95 $49.95 $19.95 $19.95 $39.95

Which of these programs represents a noninferior alternative? For those that donot, indicate which programs are clearly superior.

5.2. Consider the following multiple-objective linear program:

Maximize Z1 = 4x1 + 6x2

Maximize Z2 = —4x1 + 2x2